# Efficient computation of crystal growth using sharp interface models

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## Abstract

Untitled Efficient computation of crystal growth using sharp interface models Harald Garcke (joint work with John W. Barrett, Robert Nu¨rnberg) In order to compute crystal growth one needs to efficiently track the interface between different phases. In simple situations this results in the problem of com- puting the solution of a geometric evolution equation involving the curvature of the interface. In this talk, we review a variational formulation for such geometric evolution laws that leads to discretizations with very good mesh properties, and we indicate how these formulations can be extended to situations where the interface evolution is coupled to a bulk equation. For simplicity we state the equations for curves in the plane first and later indicate how the approach can be generalized to higher dimensions. Given a parameterization ~x(ρ, t) : I × [0, T ] → R2, I := R/Z, of the family of closed curves Γ(t) ⊂ R2, we note that the L2- and H−1-gradient flow of length, i.e. the curvature flow and the surface diffusion flow, respectively, can be written as (1) ~xt . ~ν = { κ −κss , κ ~ν = ~xss , with κ the curvature of Γ and ~ν a unit normal. Note that the formulation (1) is independent of the tangential component, ~xt . ~xs, of the velocity ~xt. However, when (1) is discretized with the help of piecewise linear finite elements, then the corresponding discrete tangential velocity is no longer arbitrary. In fact, the spa- tially discrete solutions are such that the polygonal curves Γh(t), where they are not locally flat, are equidistributed at every time t > 0. On introducing the appropriate spaces V h and V h of periodic piecewise lin- ear vector- and scalar-valued parametric finite elements, we obtain the following semidiscrete continuous-in-time approximation of (1). Given Γh(0), for t ∈ (0, T ] find Γh(t) = ~Xh(I, t), with ~Xh(t) ∈ V h, and κh(t) ∈ V h such that 〈 ~Xht , χ ~ν h〉hh − { 〈κh, χ〉hh 〈κhs , χs〉h = 0 ∀ χ ∈ V h,(2a) 〈κh ~νh, ~η〉hh + 〈 ~Xhs , ~ηs〉m = 0 ∀ ~η

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